Problem: Simplify $\cos 36^\circ - \cos 72^\circ.$
Let $a = \cos 36^\circ$ and $b = \cos 72^\circ.$  Then
\[b = \cos 72^\circ = 2 \cos^2 36^\circ - 1 = 2a^2 - 1.\]Also,
\[a = \cos 36^\circ = 1 - 2 \sin^2 18^\circ = 1 - 2 \cos^2 72^\circ = 1 - 2b^2.\]Adding these equations, we get
\[a + b = 2a^2 - 2b^2 = 2(a + b)(a - b).\]Since $a$ and $b$ are positive, $a + b \neq 0.$  We can then divide both sides by $2(a + b),$ to get
\[a - b = \boxed{\frac{1}{2}}.\]